QB 

3*75 

M»*.S58\ 
S3 


UC-NRLF 


EXCHANGE 


17  1916 


COMPUTATION  OF  THE  ORBIT  OF  PLANET 

(558) 


BY 


J.  H.  SCARBOROUGH 


SUBMITTED  FQR  THE  DEGREE  OF  DOCTOR  OF  PHILOSOPHY 

(Pn.D.)  TO  VAKDERBILT  UNIVERSITY, 

NASHVILLE,  TENN. 


MAY,  1908 


PRESS  OF 

THE  NEW  ERA  PRINTING  COMPANY 
LANCASTER,  PA. 


COMPUTATION  OF  THE  ORBIT  OF  PLANET 

(558) 


BY 

J.  H.  SCARBOROUGH 


SUBMITTED  FOR  THE  DEGREE  OF  DOCTOR  OF  PHILOSOPHY 

(PH.D.)  TO  VANDERBILT  UNIVERSITY, 

NASHVILLE,  TENN. 


MAY,  1908 


PRESS  OF 
THE  NEW  ERA  PRINTING  COMPANY 


S3 


•  -      « 


COMPUTATION  OF  THE  ORBIT  OF  PLANET  (558). 


The  small  planet  (558)  was  discovered  by  Wolf  at  Konig- 
stuhl-Heidelberg,  February  9,  1905,  and  was  reported  in  the 
Astronomische  Nachrichten,  Volume  167,  page  208,  as  follows : 

"  Photographische  Avfnahmen  von  kleinen  Planeten. 
1905.     Q,B,       M.  Z.  Kgst.       llh       51.9**,      Feb.  9,  1905. 
a  9h       47.5m 

B         +  13°       4'.0 
Gr.  12.0 

Tag.  Bewegungen,     —  0.9m  +  4'.0. 
Q,B,  und****  sind  neue  Planeten. 

M.  Wolf." 

The  value  of  Tag.  Bewegungen  as  given  in  the  above  report 
is  corrected  on  page  303  of  the  same  volume,  and  should  read, 
for  declination,  +  7'  instead  of  +  4'.  As  reported  by  the  same 
observer,  page  349  of  the  same  volume,  another  observation  on 
March  13,  1905,  gives: 


M.  Z.  Kgst. 

a 
B 


9h 
16° 


0.5m 
25.5m 
24' 


The  following  observations  made  by  Palisa  at  Wien  (k.  k. 
Sternwarte)  are  reported  in  Volume  168  of  the  Astr.  Nach. : 


1905 

M.  Z.  Wien 

aapp. 

1.  par.  A 

fiapp. 

1.  par.  A 

h       m      s 

h       m      s 

Mar.     29 

9     47     31 

9    21     16.01 

8.970 

+17     20     12.9 

0.656 

"        31 

9     37     20 

9    21      9.24 

8.955 

+17    24    37.7 

0.655 

Apr.       4 

10     17     35 

9    21     13.23 

9.265 

+17    31     50.4 

0.667 

9 

9     33     14 

9    21     50.70 

9.154 

+17    37    34.3 

0.659 

«        23 

10    37     14 

9    26    38.49 

9.491 

+17    36    23.6 

0.705 

30 

8    50    38 

9    30    29.76 

9.286 

+17    27      2.5 

0.670 

May      10 

9    27     18 

9    37    34.88 

9.465 

+17      4    15.8 

0.702 

330514 


The  places  for  the  two  Heidelberg  observations  given  on  the 
preceding  page  are  also  given  at  another  place  in  the  Astr. 
Nach.  (M.  Z.  Kgst.)  : 


Feb.  9 
Mar.  13 

h   m 
11  51.5 
11   0.5 

h   m   s 
9  47  51.19 
9  25  35.30 

+13°  4  23^50 
+16  25  34.90 

I  have  examined  the  complete  files  of  the  Astr.  Nach.  from 
the  date  of  the  discovery  of  this  planet  up  to  December  8, 
1905,  a  period  of  more  than  six  months  after  the  last  observa- 
tion given  in  Vol.  168.  So  the  above  appears  to  be  a  com- 
plete list  of  all  the  observations  made  during  the  opposition 
period  immediately  following  the  discovery. 

This  planet  was  numbered  (558),  September  26,  1905,  by 
F.  Bauschinger  of  the  Astron.  Recheninstitut,  Berlin  ;  and  this 
designation,  (558),  will  be  used  in  any  future  reference  to  it  in 
this  paper. 

The  Provisional  Elements  of  (558)  have  been  computed  by 
A.  Berberich  of  the  Astron.  Recheninstitut,  Berlin,  and  are 
published  in  Vol.  169,  p.  285,  of  the  Astr.  Nach.y  as  follows 

Epoch,  Feb.  9.5  (M.  T.  Berlin)  1905. 

M 
a> 


41° 

IT 

34".4 

314 

40 

6  .0 

144 

15 

43  .8 

8 

21 

3.0 

2 

14 

1  .0 

/*         715".481 
log  a  0  .463606. 

These  elements  are  referred  to  the  mean  equinox  of  1905.0. 

It  is  my  purpose  in  this  thesis  :  To  construct,  from  the  pro- 
visional elements  just  given,  a  provisional  ephemeris  for  the 
opposition  period  from  February  9  to  May  10,  1905.  To  cor- 
rect the  provisional  elements  by  comparing  the  observed  places, 
already  given,  with  the  computed  places  to  be  shown  in  the 
provisional  ephemeris. 


I.   COMPUTATION  OF  PROVISIONAL  EPHEMEKIS. 

(a)  Eccentric  Anomaly.  —  To  determine  the  eccentric  anom- 
aly, E,  I  used  the  transcendental  equation 

M=E-e$\nE.  (1) 

Assuming  E  =  M,  a  first  approximation  was  found ;  using  the 
first  approximation  and  repeating  the  process,  a  second  approxi- 
mation was  determined ;  the  second  approximation  was  corrected 
by  means  of  the  formula 

M-E'+e-smE' 
1-e-cosE' 

and  the  results  were  found  to  satisfy  the  given  equation  (1). 
The  results  are  given  below : 


Date,  Gr. 

M.  T. 

1" 

Approx. 

2' 

'  Approx. 

A* 

E 

Feb. 

9.5 

42° 

46 

30.81 

42° 

49' 

4.62 

4.53 

42° 

49 

9.15 

« 

13.5 

43 

35 

36.09 

43 

38 

10.30 

4.50 

43 

38 

14.80 

ti 

17.5 

44 

24 

40.31 

44 

27 

14.84 

4.42 

44 

27 

19.26 

11 

21.5 

45 

13 

43.49 

45 

16 

18.18 

4.39 

45 

16 

22.57 

14 

25.5 

46 

2 

45.59 

46 

5 

20.35 

4.32 

46 

5 

24.67 

Mar. 

1.5 

46 

51 

46.61 

46 

54 

21.29 

4.24 

46 

54 

25.53 

<« 

5.5 

47 

40 

46.56 

47 

43 

21.02 

4.17 

47 

43 

25.19 

M 

9.5 

48 

29 

45.36 

48 

32 

19.52 

4.09 

48 

32 

23.61 

|4 

13.5 

49 

18 

43.04 

49 

21 

16.75 

4.02 

49 

21 

20.77 

II 

17.5 

50 

7 

39.56 

50 

10 

12.73 

3.92 

50 

10 

16.65 

(( 

21.5 

50 

56 

34.95 

50 

59 

7.40 

3.86 

50 

59 

11.26 

{( 

25.5 

51 

45 

29.13 

51 

48 

0.82 

3.75 

51 

48 

4.57 

M 

29.5 

52 

34 

22.15 

52 

36 

52.90 

3.68 

52 

36 

56.58 

Apr. 

2.5 

53 

23 

13.95 

53 

25 

43.69 

3.56 

53 

25 

47.25 

14 

6.5 

54 

12 

4.51 

54 

14 

33.13 

3.47 

54 

14 

36.60 

II 

10.5 

55 

00 

53.93 

55 

3 

21.23 

3.37 

55 

3 

24.60 

14 

14.5 

55 

49 

42.07 

55 

52 

7.96 

3.27 

55 

52 

11.23 

14 

18.5 

56 

38 

28.93 

56 

40 

53.33 

3.18 

56 

40 

56.51 

(4 

22.5 

57 

27 

14.54 

57 

29 

37.35 

3.06 

57 

29 

40.41 

(r 

26.5 

58 

15 

58.86 

58 

18 

19.94 

2.94 

58 

18 

22.88 

II 

30.5 

59 

4 

41.91 

59 

7 

1.13 

2.84 

59 

7 

3.97 

May 

4.5 

59 

53 

23.66 

59 

55 

40.93 

2.73 

59 

55 

43.66 

it 

8.5 

60 

42 

4.06 

60 

44 

19.29 

2.63 

60 

44 

21.92 

" 

12.5 

61 

30 

43.18 

61 

32 

56.23 

2.54 

61 

32 

58.77 

All  the  values  of  E  given  in  the  above  table  have  been  veri- 
fied by  substituting  in  the  given  equation  (1). 


(6)  True  Anomaly,  Radius  Vector,  and  Longitude.  —  The 
true  anomaly,  V,  the  radius  vector,  r,  and  the  longitude  of 
the  planet  in  its  orbit,  u,  were  determined  from  the  following  : 


V 
sin  -   = 


sin  (45 


E 

sin 


V  E 

cos  -^  =  T/2a  cos  (45°  +  }</>)  •  cos  -~, 


TT  =  a)  +  a  =  458°  55'  49".8. 
The  results  are  shown  in  the  following  table  : 


Date,  Gr 

M.  T. 

V 

u 

logr 

o 

/ 

// 

o 

// 

Feb. 

9.5 

44 

21 

34.00 

359 

i 

40.00 

0.451  010 

« 

13.5 

45 

12 

3.66 

359 

52 

9.66 

0.451  180 

u 

17.5 

46 

2 

30.68 

0 

42 

36.68 

0.451  350 

ft 

21.5 

46 

52 

55.18 

1 

33 

1.18 

0.451  530 

« 

25.5 

47 

43 

17.54 

2 

23 

23.54 

0.451  706 

Mar. 

1.5 

48 

33 

37.14 

3 

13 

43.14 

0.451  886 

u 

5.5 

49 

23 

54.54 

4 

4 

0.54 

0.452  066 

« 

9.5 

50 

14 

9.10 

4 

54 

15.10 

0.452  250 

it 

13.5 

51 

4 

20.74 

5 

44 

26.74 

0.452  438 

i  ( 

17.5 

51 

54 

30.56 

6 

34 

36.56 

0.452  626 

n 

21.5 

52 

44 

36.98 

7 

24 

42.98 

0.452  818 

(i 

25.5 

53 

34 

41.14 

8 

14 

47.14 

0.453  010 

ii 

29.5 

54 

24 

42.74 

9 

4 

48.74 

0.453  204 

Apr. 

2.5 

55 

14 

41.56 

9 

54 

47.56 

0.453  400 

<t 

6.5 

56 

4 

37.64 

10 

44 

43.64 

0.453  600 

« 

10.5 

56 

54 

31.14 

11 

34 

37.14 

0.453  798 

u 

14.5 

57 

44 

21.20 

12 

24 

27.20 

0.454  004 

« 

18.5 

58 

34 

8.02 

13 

14 

14.92 

0.454  206 

« 

22.5 

59 

23 

53.88 

14 

3 

59.88 

0.454  414 

« 

26.5 

60 

13 

36.08 

14 

53 

42.08 

0.454  618 

<  < 

30.5 

61 

3 

15.00 

15 

43 

21.00 

0.454  830 

May 

4.5 

61 

52 

51.26 

16 

32 

57.26 

0.455  042 

it 

8.5 

62 

42 

25.06 

17 

22 

31.06 

0.455  250 

if 

12.5 

63 

31 

54.90 

18 

12 

0.90 

0.455  468 

(c)   The  Constants  of  Gauss.  —  The  constants  of  Gauss  were 
determined  by  means  of  the  formulas  : 


tan  N  = 


tan  i 
cos  fl ' 


cot 

A  sss  tan  H  cos  i  , 

sin  A  ' 

cot 

cos  i  •  cos  (  N  +  c) 

sin  11  cos  e 

~~  tan  ft  cos  JV'-  cos  e  ' 

bill    tf  —  -        .       -fi 

sm  ^ 

r».nt 

cos  i-sin  (^V+  e) 

sin  11  sin  e 

cin    /»  

~  ,  ;  -  -^ 

tan  11  cos  N-  sin  e  '  sm  C" 

sin  6-  sin  c-  sin  (O— 


V  ermcation  formula      tan  i  = 


sin  a  •  cos     . 
The  results  follow  : 

^1  =  234°  33'  4".32, 

^  =  146     00  18  .22, 

C=  128  43  19  .32, 
log  sin  a  =  9.998  433, 
log  sin  6=  9.981  523, 
log  sin  c  =  9.474  126. 

These  results  were  verified  by  the  verification  formula.  They 
were  used  in  the  determination  of  the  heliocentric  coordinates 
of  the  planet,  as  shown  in  the  next  section. 

(c?)  Heliocentric   Coordinates.  —  The  heliocentric  coordinates 
x,  y,  z,  were  found  from  the  following  : 

x  =  r  •  sin  a  •  sin  (  A  +  u), 
y  =  r  •  sin  b  •  sin  (5  -f  u), 
z  =  r  •  sin  c  •  sin  (O  -f  u)y 

r  being  the  radius  vector  of  the  planet,  u  the  longitude  of  the 
planet  in  its  orbit,  and  a,  6,  c,  A,  B,  C,  the  constants  of  Gauss 
already  determined. 

The  results  are  given  in  the  table  on  page  6  : 

(e)  Rectangular   Coordinates  of  the  Sun.  —  The  rectangular 


coordinates  of  the  sun,  X,  Y,  Z  (referred  to  the  mean  equinox 
of  1 905.0),  are  taken  from  the  Nautical  Almanac.  I  do  not  con- 
sider it  necessary  to  reproduce  them  here. 


Date 

X 

y 

z 

Feb. 

9.5 

—2.264  979 

+1.551  557 

+0.665  492 

13.5 

—2.290  179 

+1.519  403 

+0.658  111 

17.5 

—2.314  879 

+1.486  907 

+0.650  587 

21.5 

—2.339  139 

+1.454  153 

+0.642  938 

25.5 

—2.362  872 

+1.421  055 

+0.635  147 

Mar. 

1.5 

—2.386  117 

+1.387  669 

+0.627  221 

5.5 

—2.408  856 

+1.353  984 

+0.619  164 

9.5 

—2.431  106 

+1.320026 

+0.610  981 

13.5 

—2.452  856 

+1.285  816 

+0.602  674 

17.5 

—2.474  088 

+1.251  297 

+0.594  237 

21.5 

—2.494  806 

+1.216544 

+0.585  681 

25.5 

—2.515  006 

+1.181  528 

-  -0.577  000 

29.5 

—2.534  678 

+1.146271 

—0.568201 

Apr. 

2.5 

—2.553  835 

+1.110772 

-  -0.559  280 

6.5 

—2.572  467 

+1.075051 

-  -0.550  249 

10.5 

—2.590  550 

+1.039  100 

-  -0.541  098 

14.5 

—2.608  135 

+1.002  951 

-  -0.53  1846 

18.5 

—2.625  156 

+0.966  580 

+0.522  477 

22.5 

—2.641  663 

+0.930  022 

+0.513  003 

« 

26.5 

—2.657  577 

+0.893  256 

+0.503  420 

n 

30.5 

—2.672  994 

+0.856  328 

+0.493  741 

May 

4.5 

—2.687  850 

+0.819  213 

+0.483  959 

u 

8.5 

-2.702  149 

0.781  925 

+0.474  070 

tt 

12.5 

—2.715919 

+0.744  500 

+0.464  096 

(/)  Geocentric  Coordinates  of  the  Planet.  —  The  geocentric 
coordinates  of  the  planet,  right  ascension,  declination,  and  the 
distance  from  the  center  of  the  earth  were  determined  by  the 
formulas  : 

x  +  X  —  p  cos  S  cos  a, 


y  +  Y 
3  +  Z 


p  cos  a  sn 
p  sin  8. 


These  coordinates  were  computed  directly  for  every  four  days 
from  February  9,  1905,  to  May  12,  1905,  and  the  intermediate 
values  found  by  interpolation. 

Thus  I  have  obtained  the  desired  provisional  ephemeris 
which  is  given  in  the  table  on  next  page. 


PROVISIONAL  EPHEMERIS  OF  PLANET  (558),  FEBRUARY  9  TO  MAY  12,  1905. 


Date 

a 

S 

lOgp 

Feb. 

9.5 

h   m   s 
9  47  51.21 

13°  4  34.63 

0.264  742 

« 

10.5 

9  47   3.43 

13  11  40.53 

u 

11.5 

9  46  15.55 

13  18  46.17 

« 

12.5 

9 

13  25  51.41 

« 

13.5 

9  44  39.29 

13  32  56.09 

0.264  448 

ii 

14.5 

13  40   0,68 

u 

15.5 

9  43   3.26 

13  47   3.21 

ii 

16.5 

« 

17.5 

9  41  27.73 

14   1   3.78 

0.265261 

u 

18.5 

« 

19.5 

9  39  53.10 

14  14  54.46 

ii 

21.5 

9  38  19.85 

14  28  32.67 

0.267  181 

11 

23.5 

9  36  48.47 

14  41  56.42 

ii 

25.5 

9  35  19.36 

14  55   2.35 

0.270  147 

« 

27.5 

9  33  52.91 

15   7  47.53 

Mar. 

1.5 

9  32  29.56 

15  20  10.48 

0.274  126 

ti 

3.5 

9  31   9.80 

15  32   9.12 

ii 

5.5 

9  29  53.88 

15  43  41.36 

0.279  043 

« 

7.5 

9  28  42.06 

15  54  44.65 

« 

9.5 

9  27  34.73 

16   5  18.23 

0.284836 

ii 

10.5 

9  27   2.88 

16  10  23.42 

« 

11.5 

9  26  32.30 

16  15  20.75 

« 

12.5 

9  26   2.96 

16  20  10.13 

ii 

13.5 

9  25  34.92 

16  24  51.41 

0.291  401 

« 

14.5 

9  25   8.21 

16  29  24.41 

u 

15.5 

9  24  42.84 

16  33  49.11 

ii 

16.5 

9  24  18.82 

16  38   5.49 

«< 

17.5 

9  23  56.17 

16  42  13.42 

0.298  638 

« 

19.5 

9  23  15.03 

16  50   3.62 

« 

21.5 

9  22  39.52 

16  57  19.61 

0.306  456 

« 

23.5 

9  22   9.77 

17   4   1.03 

ii 

25.5 

9  21  45.80 

17  10   8.01 

0.314  761 

11 

26.5 

9  21  36.00 

17  12  58.59 

ii 

27.5 

9  21  27.66 

17  15  40.55 

« 

28.5 

9  21  20.78 

17  18  13.92 

11 

29.5 

9  21  15.38 

17  20  38.65 

0.323  468 

« 

30.5 

9  21  11.47 

17  22  54.70 

« 

31.5 

9  21   9.02 

17  25   2.15 

Apr. 

1.5 

9  21   8.04 

17  27   1.08 

it 

2.5 

9  21   8.54 

17  28  51.49 

0.332496 

(( 

3.5 

9  21  10.51 

37  30  33.38 

n 

4.5 

9  21  13.95 

17  32   6.85 

ii 

5.5 

9  21  18.83 

17  33  31.96 

« 

6.5 

9  21  25.16 

17  34  48.77 

0.341  774 

« 

7.5 

9  21  32.93 

17  35  57.33 

« 

8.5 

9  21  42.12 

17  36  57.71 

u 

9.5 

9  21  52.72 

17  37  49.95 

« 

10.5 

9  22   4.74 

17  38  34.11 

0.351  210 

« 

11.5 

9  22  18.17 

17  39  10.21 

<< 

12.5 

9  22  32.99 

17  39  38.34 

11 

13.5 

17  39  58.61 

8 


PROVISIONAL  EPHEMERIS  OF  PLANET  (558),  FEBRUARY  9  TO  MAY  12, 1905. 

Continued. 


Date 

a 

£ 

logp 

Apr. 

14.5 

h   m   s 
9  23   6.70 

17°  40'  1L10 

0.369  758 

ft 

15.5 

17  40  15.90 

« 

16.5 

9  23  45.73 

17  40  13.06 

<< 

17.5 

9 

17  40   2.67 

« 

18.5 

9  24  29.98 

17  39  44.72 

0.370341 

« 

19.5 

17  39  19.30 

K 

20.5 

9  25  19.33 

17  38  46.53 

it 

21.5 

9  25  45.87 

17  38   6.44 

« 

22.5 

9  26  13.61 

17  37  19.32 

0.379  923 

n 

23.5 

9  26  42.53 

17  36  25.20 

n 

24.5 

9  27  12.63 

17  35  24.09 

« 

25.5 

9  27  43.91 

17  34  15.95 

« 

26.5 

9  28  16.34 

17  33   0.82 

0.389  453 

« 

27.5 

9  28  49.95 

17  31  38.62 

« 

28.5 

9  29  24.69 

17  30   9.52 

« 

29.5 

9  30   0.54 

17  28  33.65 

<( 

30.5 

9  30  37.49 

17  26  51.10 

0.398  912 

May 

1.5 

9  31  15.52 

17  25   2.03 

« 

2.5 

9  31  54.61 

17  23   6.42 

« 

3.5 

9  32  34.75 

17  21   4.29 

« 

4.5 

9  33  15.92 

17  18  55.68 

0.408  252 

« 

5.5 

9  33  58.13 

17  16  40.58 

« 

6.5 

9  34  41.34 

17  14  19.12 

(i 

7.5 

9  35  25.53 

17  11  51.42 

(4 

8.5 

9  36  10.68 

17   9  17.57 

0.417  450 

« 

9.5 

9  36  56.77 

17   6  37.63 

(( 

10.5 

9  37  43.79 

17   3  51.69 

« 

11.5 

9  38  31.71 

17   0  59.83 

<( 

12.5 

9  39  20.52 

16  58   2.13 

0.426  481 

In  the  computation  of  the  ephemeris  just  given,  the  method 
of  differences  was  used  to  detect  errors  in  the  results.  The 
errors  thus  found  were  corrected  by  a  recomputation. 

A  portion  of  the  ephemeris  is  given  for  every  two  days,  in- 
stead of  for  every  day,  as  the  values  for  the  omitted  dates  were 
not  needed  in  determining  the  computed  coordinates  to  be  com- 
pared with  the  observed  coordinates. 

II.    To  FIND  CORRECTION  FOR  THE  PROVISIONAL 

ELEMENTS. 

(a)  Comparison  of  Computed  with  Observed  Places.  —  For 
purposes  of  comparison  the  times  of  observation  at  Heidelberg 


9 


and  at  Vienna  were  freed  from  aberration  and  reduced  to 
Greenwich  mean  time.  In  making  this  reduction  I  used  the 
constants  for  the  K.  K.  Universitats-Sternwarte  in  Wien,  as 
given  in  the  Annalen  der  K.  K.  Sternwarte  zu  Wien,  Vol.  XV, 
p.  113.  I  give  them  below  : 

Geocentric  latitude  48°      2'  55".4 

Longitude  (from  Greenwich)    —    lh       5m  21'  .49,  or 

- 16°     20'  22".3 
Log  of  dist.  from  center  of  earth  9.999  211 

The  corresponding  constants  for  Heidelberg  are  : 

Geocentric  latitude  49°        23'  2".55 

Longitude  (from  Gr.)         —    Oh         34m         48'  .5 
Log  dist.  from  center  of  earth  9.999  075 

Using  the  ephemeris  given  in  Part  I  of  this  thesis,  I  found  by 
interpolation  the  coordinates  of  the  planet  for  the  Greenwich 
mean  time  of  observation  corrected  for  aberration.  The  results 
are  given  in  the  table  below.  The  coordinates  given  in  the 
table  are  the  true  coordinates  of  the  place  of  the  planet  referred 
to  the  center  of  the  earth. 

COMPUTED  a  AND  6  AT  TIMES  OF  OBSERVATION. 


Date  (Gr.  M.  T.  Freed 
from  Abbr.) 

a 

a 

Feb.    9^4598 

h   m   s 
9  47  53.13 

13°  4  17'i2 

Mar.   13.4211 

9  25  37.08 

16  24  29.52 

"    29.3505 

9  21  16.09 

17  20  17.66 

"    31.3432 

9  21   9.31 

17  24  42.74 

Apr.    4.3710 

9  21  13.62 

17  31  55.26 

"     9.3398 

9  21  50.93 

17  37  42.12 

"    23.3832 

9  26  39.09 

17  36  31.88 

"    30.3086 

9  30  30.23 

17  27  11.11 

May   10.3334 

9  37  35.89 

17   4  19.75 

For  convenenience  in  comparison  with  the  computed  coordi- 
nates, the  observed  coordinates  were  corrected  for  parallax  in 
right  ascension  and  declination,  and  reduced  to  the  mean  equinox 
of  1905.0. 


10 


The  parallax  factors,  given  on  page  2,  must  each  be  divided 
by  the  distance  from  the  planet  to  the  earth  at  a  given  date,  in 
order  to  obtain  the  correction  for  parallax.  The  parallax  fac- 
tors were  not  given  for  the  two  Heidelberg  observations,  but  I 
computed  them  by  the  usual  formulas. 

The  following  table  gives  the  corrections  to  be  applied ;  and 
the  arrangement  is  such  that : 

"  I "  is  correction  for  parallax  in  right  ascension. 

"  II "  is  correction  in  right  ascension  for  mean  equinox  of 
1905.0. 

"  III "  is  correction  for  parallax  in  declination. 

"  IV "  is  correction  in  declination  for  mean  equinox  of 
1905.0. 


Date 

I 

II 

III 

IV 

Feb. 

9.4598 

-08.035 

+0.067 

+2/./83 

+4.94 

Mar. 

13.4211 

+0.047 

—0.128 

+4.79 

+6.32 

« 

29.3505 

+0.044 

—0.228 

+2.15 

+6.86 

it 

31.3432 

+0.043 

—0.240 

+2.13 

+6.88 

Apr. 

4.3710 

+0.085 

—0.239 

+2.14 

+6.85 

ft 

9.3398 

+0.064 

-0.261 

+2.05 

+7.08 

« 

23.3832 

+0.128 

-0.356 

+2.10 

+7.56 

« 

30.3086 

+0.077 

—0.405 

+1.87 

+7.67 

May 

10.3334 

+0.111 

—0.484 

+1.98 

+8.09 

These  corrections  were  applied  to  the  apparent  coordinates 
given  on  page  2.  The  results  are  given  in  the  table  below. 
The  values  of  a  and  8  in  this  table  are  the  "  observed  "  values 
after  the  corrections  are  applied.  In  the  column  marked 
"  0  —  C,  a  "  the  difference  between  the  "  observed  "  and  "  com- 
puted "  values  of  a  are  given,  the  sign  being  in  the  order  of 
observed  minus  computed.  In  the  column  marked  "  0  —  (7,  5  " 
the  differences  between  the  observed  and  computed  values  of 
declination  are  given. 

Computed  values  are  given  on 

It  will  be  noticed  that  the  two  Heidelberg  observations  give 
much  larger  differences  than  the  others.  This  would  seem  to 
indicate  that  there  is  some  error  not  accounted  for.  The  re- 


11 


Date 


0—  C,  a 
Aa 


o-c;  s 

AS 


Feb. 

9.4598 

h 
9 

m 
47 

51.22 

13° 

4 

31.27 

—1.91 

+13.65 

Mar. 

13.4211 

9 

25 

35.22 

16 

24 

46.01 

—1.86 

+16.49 

it 

29.3505 

9 

21 

15.83 

17 

20 

21.91 

—0.26    +  4.25 

« 

31.3432 

9 

21 

9.04 

17 

24 

46.71 

—0.27    +  3.97 

Apr. 

4.3710 

9 

21 

13.08 

17 

31 

59.39 

—0.54 

+  4.13 

ft 

9.3398 

9 

21 

50.51 

17 

37 

43.43 

—0.42 

+  1.31 

« 

23.3832 

9 

26 

38.26 

17 

36 

33.26 

—0.83 

+  1.38 

K 

30.3086 

9 

30 

29.43 

17 

27 

12.04 

—0.80 

+  0.93 

May 

10.3334 

9 

37 

34.52 

17 

4 

25.87 

—1.37 

+'.o;i2 

ports  of  these  two  observations  given  in  the  Astr.  Nach.  give 
the  time  only  to  the  tenths  of  a  minute.  This  is  not  sufficient 
to  account  for  the  large  differences  referred  to ;  yet,  as  a  part  of 
the  report  gives  only  approximate  values,  it  is  possible  that  the 
coordinates  given  are  only  approximate  values  rather  than  care- 
fully measured  results.  Not  having  time  to  communicate  with 
the  observer  at  Heidelberg,  I  decided  to  leave  out  these  two 
observations  and  base  the  remainder  of  my  computations  on 
the  other  observations. 

(6)  Normal  Places.  —  Omitting  the  two  Heidelberg  observa- 
tions, I  collected  the  others  into  three  groups : 

Group  I,  observations  of  March  29  and  31. 

Group  II,  observations  of  April  4,  9  and  23. 

Group  III,  observations  of  April  30  and  May  10. 

Taking  the  average  of  the  differences  for  each  group  as  the 
correction  for  the  computed  place  at  the  beginning  of  the  day 
nearest  the  mean  date,  we  have  the  following  results  : 


Average 

Normal  Places 

Group 

Date 

Aa 

AJ 

a 

S 

I 

Mar.  30.5 

S 
-0.27 

n 
+4.11 

h       m       s 
9     21     11.20 

17°  22    58.81 

II 
III 

Apr.  14.5 
May     5.5 

—0.60 
—1.08 

+2.27 
+3.82 

9     23       6.10 
9    33    57.05 

17    40     13.37 
17     16    44.40 

(c)  Computation  of  Corrections  of  the  Elements.  —  To  com- 
pute the  corrections  for  the  elements,  I  first  referred  the  position 
of  the  planet  to  the  ecliptic,  and  computed  the  latitude  and 


12 

longitude  at  the  times  given  in  the  table  for  normal  places. 
After  taking  account  of  the  latitude  of  the  sun,  and  computing 
and  checking  in  the  usual  manner,  I  found  the  coordinates 
to  be: 


Date 

Latitude 

Longitude 

Mar. 

30.5 

1° 

48     10.348 

137°  16     20!83 

Apr. 

14.5 

2 

13    00.160 

137    37      7.46 

May 

5.5 

2 

39    31.170 

140     12      0.05 

It  will  be  noticed  that  the  latitudes  of  the  planet  are  small, 
and  the  angle  between  two  of  the  radii  vectores  still  smaller. 
Slight  errors  in  observations,  or  in  certain  approximations 
would  have  considerable  effect.  Moreover,  after  continuing 
the  work  at  some  length,  I  found  that  in  certain  portions  of 
the  work  it  was  necessary  to  carry  the  computation  beyond 
seven  decimal  places  ;  and  I  did  not  have  a  complete  table  of 
more  than  seven  places.  So,  after  spending  considerable  time 
and  labor,  I  found  that  it  was  necessary  to  change  my  plan, 
and  refer  the  planet  to  the  equator  as  the  plane  of  reference. 

I  then  used  the  differential  method,  and  found  the  changes 
that  would  be  produced  in  the  elements  by  the  variations  in 
right  ascension  and  decimation  given  on  page  20.  By  differ- 
entiating the  equations  involving  the  elements  and  the  coordi- 
nates considered  as  variables,  formulas  were  obtained  giving 
the  relations  between  the  variations  in  the  elements  and  the 
variations  in  the  coordinates.  These  differentiations,  after 
some  reduction  to  more  convenient  forms,  give, 

1  °  For  variation  in  right  ascension, 

>  da  ,,  *  da  da  da  A 

cos  o  -r-  ATT  -f  cos  o  -77^1!  -f-  cos  o  -yv  A^  +  cos  o  -jj  Ad> 
dir  d£l  di  d<f) 

-j-  cos  8  -7^  Alf  -f  cos  8  ~=-  Au  =  cos  8  •  Aa. 
dM  dfji 

2°  For  variation  in  declination, 

dS  A  d8  A          d8  A        d8  d8  d8  A 

-j-  ATT  -f  -^  All  +  -y.Ai  +  -r.  A<£  -f  -^  Alf-f  -,-  A/*  =  AS. 

d7r  d£l  di  d<j>  dM  dp 


13 

In  order  to  obtain  the  six  equations  between  the  variations 
of  the  six  elements,  I  took  the  combined  results  of  the  vari- 
ations of  the  coordinates  given  for  the  normal  places  ;  the  three 
changes  in  right  ascension  giving  me  three  equations,  and  the 
three  changes  in  declination  giving  the  other  three. 

I  might  have  taken  each  observation  separately,  and  from  the 
seven  observations  obtained  fourteen  equations,  and  then  com- 
bined these  into  six  equations.  By  either  method  of  obtaining 
the  six  equations,  all  the  observations  available  are  made 
use  of. 

The  equations  of  condition  for  finding  the  values  of  the  vari- 
ations of  the  elements  were  thus  found  to  be  : 

+  1.275012  Air— 0.045347  AQ+0.035849  Ai+1. 895022  A^-f  1.347753  AJf—  1.752088    Aju=  3".8650 


—0.196062 

—0.187308 

+0.213273 

-0.250601 

-0.209490 

+13.394215 

=  4  .110 

+  1.161339 

—0.041779 

+0.044644 

+1.778783 

+1.225739 

+  5.436958 

=  8  .5754 

—0.175604 

—0.171191 

+0.262144 

-0.229251 

-0.187798 

+13.018983 

=  2  .270 

+  1.025810 

—0.038474 

+0.057220 

+1.654673 

+1.078990 

+20.764357 

=15  .469 

-0.164980 

—0.149625 

+0.313806 

-0.234041 

-0.175896 

+  9.383532 

=  3  .820 

The  second  member  of  each  of  the  odd-numbered  equations 
above  is  cos  8  •  Aa,  while  the  second  member  of  each  of  the 
others  is  A8.  The  second  member  in  each  is  expressed  in  sec- 
onds of  arc. 

To  verify  the  computations  of  the  coefficients  in  the  above 
equations,  I  assumed  a  small  arbitrary  change  in  each  element, 
namely  : 

Let  ATT  =  -  20" 


-10 
Ai  =  -f  10 
A<£  =  +  10 
AJf  =  +  10 
A/A  =  +  00.91 

Applying  these  changes  to  the  provisional  elements,  and 
using  the  elements  thus  corrected,  I  computed  the  right 
ascension  of  the  planet  for  April  14.5  and  compared  it  with 


14 

the  right  ascension  already  computed  from  the  elements  before 
the  changes  were  applied.     I  found  the  difference  to  be, 

Act  =  8.48". 

I  also  found  the  value  of  Act  directly  from  the  third  equa- 
tion in  the  list,  using  the  same  date,  and  the  agreement  was 
close  enough  to  indicate  that  the  computation  of  the  coefficients 
in  this  equation  is  correct.  As  the  computation  of  the  places 
in  this  verification  is  rather  long,  I  did  not  apply  it  to  all  the 
equations.  However,  it  is  very  probable  that  if  an  error  should 
occur  in  the  computation  of  the  coefficients  of  any  one  of  the 
equations,  it  would  affect  the  coefficients  in  the  others  also. 

The  solution  of  the  six  equations  given  above,  involving  the 
variations  of  the  six  elements,  gives 

AJf  =  +  3'   19".49 
A<£  =  +  V  46".15 
Ai    =  +  4'.80 
AH=  +   8".33 
ATT  =  -   6'  5".99 
A/i  =  -    0".001386. 

I  found  Act  from  the  formula,  A  =ct  —  f  -    , 
the  result  being,          Aa  =  0.00000376. 

Applying  the  corrections  just  found  to  the  provisional  ele- 
ments, we  have  the  following  : 

M=  41°  20'  53".89 
7r=  98  49  43  .81 
ft  =  144  15  52  .13 
i=  8  21  8  .80 
<£  =  2  15  47  .15 
H=  715  .4796 

Note  that  TT,  used  above,  is  equal  to  fl  -f  &>  (used  on  page  3). 


15 

The  epoch  is  the  same  as  on  page  3,  and  the  elements  are 
referred  to  the  mean  equinox  of  1905.0. 

These  are  the  corrected  elements  sought. 

The  approximate  time  of  the  second  opposition  is  May  17, 
1906,  the  time  of  first  opposition  being  February  4,  1905. 

From  the  mean  daily  motion  as  corrected,  the  sidereal  period 
is  found  to  be  1811.37  days ;  the  synodic  period  can  be  readily 
found  by  comparing  with  the  period  of  the  earth — it  is  nearly 
457  days. 

The  perturbations  of  Jupiter  and  Saturn  have  not  been 
taken  into  account.  This  topic  would  of  itself  form  an  inter- 
esting one  for  a  thesis. 

Conclusion. —  I  cannot  hope  that  the  corrected  elements  here 
given  are  free  from  error.  The  best  results  of  computations 
based  upon  the  observations  of  one  opposition  only,  cannot  in 
any  case  give  us  results  altogether  free  from  error.  The  short 
period  over  which  the  available  observations  extend,  is  such  as 
to  prevent  results  altogether  accurate.  Some  other  difficulties 
have  already  been  pointed  out. 

I  selected  this  planet  from  a  large  list  of  newly  discovered 
asteroids  after  examining  all  the  available  observations  of  each 
one.  This  afforded  me  the  best  material  at  hand  for  the  work 
I  desired  to  accomplish,  for  the  observations  were  more  numerous 
and  extended  over  a  longer  period  than  those  of  any  other  one 
in  the  list. 

The  fact  that  all  computation  and  checking  of  results  had  to 
be  done  by  one  person  has  made  the  task  much  longer  than  I 
had  anticipated.  However,  I  hope  to  be  able  at  some  future 
time  to  carry  the  work  forward  to  other  oppositions,  as  they 
are  observed,  and  thus  make  still  further  corrections  of  the 
elements.  We  cannot  hope  for  results  of  extreme  accuracy, 
without  carrying  the  work  forward  through  several  oppositions. 

In  the  prosecution  of  the  work,  I  cannot  claim  originality  as 
to  the  theory  and  methods  used.  In  fact,  the  theory  of  the 


16 

computation  of  orbits  of  bodies  moving  about  the  sun  as  given 
by  Gauss  in  his  "  Theoria  Motus  "  a  century  ago,  is  the  basis 
of  all  the  work  of  the  computation  of  orbits  by  the  astronomers 
of  the  present  day. 

I  have  consulted  freely,  and  used  formulas,  from  the 
following : 

"  Theoria  Motus,"  by  Gauss,  translation  by  Davis. 

"  Lecons  sur  la  Determination  des  Orbits,"  par  Tisserand, 
arranged  by  Perchot. 

"  Les  Orbits  des  Cometes  et  Planetes,"  by  D'Oppolzer, 
French  translation. 

"Tafeln  zur  Theoretischen  Astronomic,"  by  Bauschinger. 

"  Theoretical  Astronomy,"  by  Watson. 

Astronomische  Nachrichten,  complete   files   to  December, 
1905. 

I  am  under  special  obligation  to  Wm.  J.  Vaughn,  LL.D., 
head  of  the  Department  of  Mathematics  and  Astronomy, 
Vanderbilt  University,  for  his  many  valuable  suggestions,  and 
also  for  the  free  use  of  his  excellent  private  library. 

I  am  also  under  special  obligation  to  Professor  D.  T.  Wilson, 
of  the  Case  School  of  Applied  Science  (Cleveland,  O.),  who  has 
kindly  verified  most  of  the  computations  in  this  thesis. 


THIS  BOOK  IS  DUE  ON  THE  LAST  DATE 
STAMPED  BELOW 


AN  INITIAL  FINE  OF  25  CENTS 

WILL  BE  ASSESSED  FOR  FAILURE  TO  RETURN 
THIS  BOOK  ON  THE  DATE  DUE.  THE  PENALTY 
WILL  INCREASE  TO  SO  CENTS  ON  THE  FOURTH 
DAY  AND  TO  $I.OO  ON  THE  SEVENTH  DAY 
OVERDUE. 


J 


LD  21-100TO-12, '43  (8796s) 


Gay  lord  Bros. 

Makers 
Syracuse,  N.  Y. 

PAT.  JAN.  21,  1908 


3  3 


t-± 


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